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Annals of Mathematics


Determination of the
algebraic relations among
special Γ-values in positive
characteristic


By Greg W. Anderson, W. Dale Brownawell, and
Matthew A. Papanikolas

Annals of Mathematics, 160 (2004), 237–313
Determination of the algebraic relations
among special Γ-values in positive
characteristic
By Greg W. Anderson, W. Dale Brownawell
∗
, and Matthew A. Papanikolas
Abstract
We devise a new criterion for linear independence over function fields. Us-
ing this tool in the setting of dual t-motives, we find that all algebraic relations
among special values of the geometric Γ-function over F
q
[T ] are explained by
the standard functional equations.
Contents
1. Introduction
2. Notation and terminology
3. A linear independence criterion
4. Tools from (non)commutative algebra

5. Special functions
6. Analysis of the algebraic relations among special Π-values
References
1. Introduction
1.1. Background on special Γ-values.
1.1.1. Notation. Let F
q
be a field of q elements, where q isapowerofa
prime p. Let A := F
q
[T ] and k := F
q
(T ), where T is a variable. Let A
+
⊂ A
be the subset of monic polynomials. Let |·|
∞
be the unique valuation of k for
which |T |
∞
= q. Let k
∞
:= F
q
(( 1 /T )) be the |·|
∞
-completion of k, let k
∞
be
an algebraic closure of k

∞
, let C
∞
be the |·|
∞
-completion of k
∞
, and let
¯
k be
the algebraic closure of k in C
∞
.
∗
The second author was partially supported by NSF grant DMS-0100500.
238 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
1.1.2. The geometric Γ-function. In [Th], Thakur studied the geometric
Γ-function over F
q
[T ],
Γ(z):=
1
z

n∈A
+

1+
z
n


−1
(z ∈ C
∞
),
which is a meromorphic function on C
∞
. Notably, it satisfies several natural
functional equations, which are the analogues of the translation, reflection and
Gauss multiplication identities satisfied by the classical Euler Γ-function, and
to which we refer as the standard functional equations (see §5.3.5).
1.1.3. Special Γ-values and the fundamental period of the Carlitz module.
We define the set of special Γ-values to be
{Γ(z) | z ∈ k \ (−A
+
∪{0})}⊂k
×
∞
.
Up to factors in k
×
a special Γ-value Γ(z) depends only on z modulo A.In
connection with special Γ-values it is natural also to consider the number
 := T
q−1
√
−T
∞

i=1


1 − T
1−q
i

−1
∈ k
∞

q−1
√
−T

×
where
q−1
√
−T is a fixed (q − 1)
st
root of −T in C
∞
. The number  is the
fundamental period of the Carlitz module (see §5.1) and hence deserves to be
regarded as the F
q
[T ]-analogue of 2πi. The transcendence of  over k was
first shown in [Wa]. (See §3.1.2 for a new proof.) Our goal in this paper to
determine all Laurent polynomial relations with coefficients in
¯
k among special

Γ-values and .
1.1.4. Transcendence of special Γ-values. For all z ∈ A the value Γ(z),
when defined, belongs to k. However, it is known that for all z ∈ k\A the value
Γ(z) is transcendental over k. A short history of this transcendence result is as
follows. Isolated results on the transcendence of special Γ-values were obtained
in [Th]; in particular, it was observed that when q = 2, all values Γ(z) with
z ∈ k \ A are
¯
k-multiples of the Carlitz period . The first transcendence
result for a general class of values of the Γ-function was obtained in [Si a].
Namely, Sinha showed that the values Γ(
a
f
+ b) are transcendental over k for
all a, f ∈ A
+
and b ∈ A such that deg a<deg f. Sinha’s results were obtained
by representing the Γ-values in question as periods of t-modules defined over
¯
k and then invoking a transcendence criterion of Gelfond-Schneider type from
[Yu a]. Subsequently all the values Γ(z) for z ∈ k \ A were represented in
[BrPa] as periods of t-modules defined over
¯
k and thus proved transcendental.
1.1.5. Γ-monomials and the diamond bracket criterion. An element of
the subgroup of C
×
∞
generated by  and the special Γ-values will for brevity’s
sake be called a Γ-monomial. By adapting the Deligne-Koblitz-Ogus criterion

ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
239
[De] to the function field setting along lines suggested in [Th], we have at our
disposal a diamond bracket criterion (see Corollary 6.1.8) capable of deciding
in a mechanical way whether between a given pair of Γ-monomials there exists
a
¯
k-linear relation explained by the standard functional equations. We call the
two-term
¯
k-linear dependencies thus arising diamond bracket relations.
1.1.6.Cautionary example. In order to deduce certain
¯
k-linear relations
between Γ-monomials from the standard functional equations, root extraction
cannot be avoided. Consider the following example concerning the classical
Γ-function taken from [Da]. The relation
Γ

4
15

Γ

1
5

Γ

1

3

Γ

2
15

=




































Γ
(
1
5
)
Γ
(
1
15
)
Γ
(
2
5
)
Γ
(

11
15
)
×
Γ
(
2
5
)
Γ
(
2
15
)
Γ
(
4
5
)
Γ
(
7
15
)
×
Γ
(
1
15
)

Γ
(
4
15
)
Γ
(
7
15
)
Γ
(
2
3
)
Γ
(
13
15
)
Γ
(
1
3
)
×
Γ
(
4
15

)
Γ
(
11
15
)
Γ
(
1
3
)
Γ
(
2
3
)
×
Γ
(
1
5
)
Γ
(
4
5
)
Γ
(
2

15
)
Γ
(
13
15
)
=3
−
1
5
5
1
12

sin
π
3
· sin
2π
15
sin
4π
15
· sin
π
5
confirms the Deligne-Koblitz-Ogus criterion but decomposes into instances of
the standard functional equations only after the terms are squared. The results
of [Ku b] imply the existence of such peculiar examples in great abundance.

See [Da] for a method by which essentially all such examples can be constructed
explicitly. The analogous phenomena occur in the function field situation. For
a discussion of the latter, see [BaGeKaYi]. For a simple example in the case
q = 3, which was in fact discovered before all the others mentioned in this
paragraph, see [Si b, §4].
1.1.7. Linear independence. It was shown in [BrPa] that the only relations
of
¯
k-linear dependence among 1, , and special Γ-values are those following
from the diamond bracket relations. This result was obtained by carefully
analyzing t-submodule structures and then invoking Yu’s powerful theorem of
the t-Submodule [Yu c].
1.2. The main result. We prove:
Theorem 1.2.1 (cf. Theorem 6.2.1). A set of Γ-monomials is
¯
k-linearly
dependent exactly when some pair of Γ-monomials is. Pairwise
¯
k-linear
(in)dependence of Γ-monomials is entirely decided by the diamond bracket cri-
terion.
In other words, all
¯
k-linear relations among Γ-monomials are
¯
k-linear com-
binations of the diamond bracket relations. The theorem has the following
implication concerning transcendence degrees:
240 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
Corollary 1.2.2 (cf. Corollary 6.2.2). For al l f ∈ A

+
of positive de-
gree, the extension of
¯
k generated by the set
{}∪

Γ(x)




x ∈
1
f
A \ ({0}∪−A
+
)

is of transcendence degree 1+
q−2
q−1
· #(A/f)
×
over
¯
k.
In fact the corollary is equivalent to the theorem (see Proposition 6.2.3).
1.3. Methods. We outline the proof of Theorem 1.2.1, emphasizing
the new methods introduced here, and compare our techniques to those used

previously.
1.3.1. A new linear independence criterion. We develop a new method
for detecting
¯
k-linear independence of sets of numbers in
k
∞
, culminating in a
quite easily stated criterion. Let t be a variable independent of T . Given f =

∞
i=0
a
i
t
i
∈ C
∞
[[ t]] and n ∈ Z, put f
(n)
:=

∞
i=0
a
q
n
i
t
i

and extend the operation
f → f
(n)
entrywise to matrices. Let E⊂
¯
k[[ t]] be the subring consisting of power
series

∞
i=0
a
i
t
i
such that [k
∞
({a
i
}
∞
i=0
):k
∞
] < ∞ and lim
i→∞
i

|a
i
|

∞
=0.
We now state our criterion (Theorem 3.1.1 is the verbatim repetition; see also
Proposition 4.4.3):
Theorem 1.3.2. Fix a matrix Φ=Φ(t) ∈ Mat
×
(
¯
k[t]) such that det Φ
is a polynomial in t vanishing (if at all) only at t = T . Fix a (column) vector
ψ = ψ(t) ∈ Mat
×1
(E) satisfying the functional equation ψ
(−1)
=Φψ. Evaluate
ψ at t = T, thus obtaining a (column) vector ψ(T ) ∈ Mat
×1

k
∞

. For every
(row ) vector ρ ∈ Mat
1×
(
¯
k) such that ρψ(T )=0there exists a (row ) vector
P = P(t) ∈ Mat
1×
(

¯
k[t]) such that P (T )=ρ and Pψ =0.
In other words, in the situation of this theorem, every
¯
k-linear relation
among entries of the specialization ψ(T ) is explained by a
¯
k[t]-linear relation
among entries of ψ itself.
1.3.3. Dual t-motives. The category of dual t-motives (see §4.4) pro-
vides a natural setting in which we can apply Theorem 1.3.2. Like t-motives in
[An a], dual t-motives are modules of a certain sort over a certain skew polyno-
mial ring. From a formal algebraic perspective dual t-motives differ very little
from t-motives, and consequently most t-motive concepts carry over naturally
to the dual t-motive setting. In particular, the concept of rigid analytic triv-
iality carries over (see §4.4). Crucially, to give a rigid analytic trivialization
of a dual t-motive is to give a square matrix with columns usable as input to
Theorem 1.3.2 (see Lemma 4.4.12).
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
241
1.3.4. Position of the new linear independence criterion with respect to
Yu ’s Theorem of the t-Submodule. We came upon Theorem 1.3.2 in the pro-
cess of searching for a t-motivic translation of Yu’s Theorem of the t-Submodule
[Yu c]. Our discovery of a direct proof of Theorem 1.3.2 was a happy accident,
but it was one for which we were psychologically prepared by close study of
the proof of Yu’s theorem.
Roughly speaking, the points of view adopted in the two theorems cor-
respond as follows. If H = Hom(G
a
,E) is the dual t-motive defined over

¯
k corresponding canonically to a uniformizable abelian t-module E defined
over
¯
k, and Ψ = Ψ(t) is a matrix describing a rigid analytic trivialization of H
as in Lemma 4.4.12, then it is possible to express the periods of E in a natural
way as
¯
k-linear combinations of entries of Ψ(T )
−1
and vice versa.Thusitbe-
comes at least plausible that Theorem 1.3.2 and Yu’s theorem provide similar
information about
¯
k-linear independence. A detailed comparison of the two
theorems is not going to be presented here; indeed, such has yet to be worked
out. But we are inclined to believe that at the end of the day the theorems
differ insignificantly in terms of ability to detect
¯
k-linear independence.
In any case, it is clear that both theorems are strong enough to handle the
analysis of
¯
k-linear relations among Γ-monomials. Ultimately Theorem 1.3.2
is our tool of choice just because it is the easier to apply. Theorem 1.3.2 allows
us to carry out our analysis entirely within the category of dual t-motives,
which means that we can exclude t-modules from the picture altogether at a
considerable savings of labor in comparison to [Si a] and [BrPa].
1.3.5. Linking Γ-monomials to dual t-motives via Coleman functions.In
order to generalize beautiful examples in [Co] and [Th], solitons over F

q
[T ] were
defined and studied in [An b]. In turn, in order to obtain various results on
transcendence and algebraicity of special Γ-values, variants of solitons called
Coleman functions were defined and studied in [Si a] and [Si b].
We present in this paper a self-contained elementary approach to Coleman
functions producing new simple explicit formulas for them (see §5, §6.3). From
the Coleman functions we then construct dual t-motives with rigid analytic
trivializations described by matrices with entries specializing at t = T to
¯
k-
linear combinations of Γ-monomials (see §6.4), thus putting ourselves in a
position where Theorem 1.3.2 is at least potentially applicable.
Our method for attaching dual t-motives to Coleman functions is straight-
forwardly adapted from [Si a]. But our method for obtaining rigid analytic
trivializations is more elementary than that of [Si a] because the explicit for-
mulas for Coleman functions at our disposal obviate sophisticated apparatus
from rigid analysis.
1.3.6. Geometric complex multiplication. The dual t-motives engendered
by Coleman functions are equipped with extra endomorphisms and are exam-
242 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
ples of dual t-motives with geometric complex multiplication, GCM for short
(see §4.6). We extend a technique developed in [BrPa] for analyzing t-modules
with complex multiplication to the setting of dual t-motives with GCM, dub-
bing the generalized technique the Dedekind-Wedderburn trick (see §4.5). We
determine that rigid analytically trivial dual t-motives with GCM are semi-
simple up to isogeny. In fact each such object is isogenous to a power of a
simple dual t-motive.
1.3.7. End of the proof. Combining our general results on the structure
of GCM dual t-motives with our concrete results on the structure of the dual

t-motives engendered by Coleman functions, we can finally apply Theorem 1.3.2
(in the guise of Proposition 4.4.3) to rule out all
¯
k-linear relations among
Γ-monomials not following from the diamond bracket relations (see §6.5), thus
proving Theorem 1.2.1.
1.4. Comments on the classical case. In the classical situation various
people have formed a clear picture about what algebraic relations should hold
among special Γ-values. Those ideas stimulated our interest and guided our
intuition in the function-field setting. We discuss these ideas in more detail
below.
1.4.1. Temporary notation and terminology. For the duration of §1.4,
let Γ(s) be the classical Γ-function, call {Γ(s) | s ∈ Q \Z
≤0
} the set of special
Γ-values, and let a Γ-monomial be any element of the subgroup of C
×
generated
by the special Γ-values and 2πi.
1.4.2. Rohrlich’s conjecture. Rohrlich in the late 1970’s made a con-
jecture which in rough form can be stated thus: all multiplicative algebraic
relations among special Γ-values and 2πi are explained by the standard func-
tional equations. See [La b, App. to §2, p. 66] for a more precise formulation
of the conjecture in the language of distributions. In language very similar to
that we have used above, Rohrlich’s conjecture can also be formulated as the
assertion that the Deligne-Koblitz-Ogus criterion for a Γ-monomial to belong
to
Q
×
is not only sufficient, but necessary as well.

1.4.3. Lang’s conjecture. Lang subsequently strengthened Rohrlich’s con-
jecture to a conjecture which in rough form can be stated thus: all polynomial
algebraic relations among special Γ-values and 2πi with coefficients in
Q are
explained by the standard functional equations. See [La b, loc. cit.] for a for-
mulation of this conjecture in the language of distributions. In language very
similar to that we have used above, Lang’s conjecture can also be formulated
as the assertion that all
Q-linear relations among Γ-monomials follow linearly
from the two-term relations provided by the Deligne-Koblitz-Ogus criterion.
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
243
Yet another formulation of Lang’s conjecture is the assertion that for every
integer n>2 the transcendence degree of the extension of
Q generated by the
set {2πi}∪

Γ(x)


x ∈
1
n
Z \ Z
≤0

is equal to 1 + φ(n)/2, where φ(n) is Euler’s
totient. In fact, as is underscored by the direct analogy between the numbers
1+φ(n)/2=1+


1 −
1
#Z
×

· #(Z/n)
×
and
1+
q −2
q −1
· #(A/f)
×
=1+

1 −
1
#A
×

· #(A/f)
×
,
Corollary 1.2.2 is the precise analogue of the last version of Lang’s conjecture
mentioned above.
1.4.4. Evidence in the classical case. There are very few integers
n>1 such that all Laurent polynomial relations among elements of the set
{2πi}∪

Γ


1
n

, ,Γ

n−1
n

with coefficients in
Q can be ruled out save those
following from the two-term relations provided by the Deligne-Koblitz-Ogus
[De] criterion, to wit:
• n = 2 (Lindemann 1882, since Γ(1/2) =
√
π).
• n =3, 4 (Chudnovsky 1974, cf. [Wal]).
The only other evidence known for Lang’s conjecture is indirect, and it is
contained in a result of [WoW¨u]: all
Q-linear relations among the special beta
values
B(a, b)=
Γ(a)Γ(b)
Γ(a + b)
(a, b ∈ Q, a,b,a+ b ∈ Z
≤0
)
and 2πi follow from the two-term relations provided by the Deligne-Koblitz-
Ogus criterion.
1.5. Acknowledgements. The authors thank Dinesh Thakur for helpful

conversations and correspondence. The second and third authors would like
to thank the Erwin Schr¨odinger Institute for its hospitality during some of the
final editorial work.
2. Notation and terminology
2.1. Table of special symbols.
T, t, z := independent variables
F
q
:= a field of q elements
k := F
q
(T )
|·|
∞
:= the unique valuation of k such that |T |
∞
= q
244 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
k
∞
:= F
q
(( 1 /T )) = the |·|
∞
-completion of k
k
∞
:= an algebraic closure of k
∞
C

∞
:= the |·|
∞
-completion of k
∞
¯
k := the algebraic closure of k in C
∞

T := a fixed choice in
¯
k ofa(q − 1)
st
root of −T
C
∞
{t} := the subring of the power series ring C
∞
[[ t]] consisting of
power series convergent in the “closed” unit disc |t|
∞
≤ 1
#S := the cardinality of a set S
Mat
r×s
(R) := the set of r by s matrices with entries in a ring or module R
R
×
:= the group of units of a ring R with unit
GL

n
(R) := Mat
n×n
(R)
×
, where R is a ring with unit
1
n
:= the n by n identity matrix
A := F
q
[T ]
deg := (a → degree of a in T ):A → Z ∪{−∞}
A
+
:= the set of elements of A monic in T
D
N
:=

N−1
i=0
(T
q
N
− T
q
i
) ∈ A
+

Res :=


i
a
i
T
i
→ a
−1

: k
∞
→ F
q
2.2. Twisting. Fix n ∈ Z. Given a formal power series f =

∞
i=0
a
i
t
i
∈
C
∞
[[ t]] we define the n-fold twist by the rule f
(n)
:=


∞
i=0
a
q
n
i
t
i
. The n-fold
twisting operation is an automorphism of the power series ring C
∞
[[ t]] stabi-
lizing various subrings, e. g.,
¯
k[[ t]] ,
¯
k[t], and C
∞
{t}. More generally, for any
matrix F with entries in C
∞
[[ t]] we define the n-fold twist F
(n)
by the rule

F
(n)

ij
:= (F

ij
)
(n)
. In particular, for any matrix X with entries in C
∞
we
have

X
(n)

ij
=(X
ij
)
q
n
. The n-fold twisting operation commutes with matrix
addition and multiplication.
2.3. Norms. For any matrix X with entries in C
∞
we put |X|
∞
:=
max
ij
|X
ij
|
∞

.Now


X
(n)


∞
= |X|
q
n
∞
for all n ∈ Z and
|U + V |
∞
≤ max(|U|
∞
, |V |
∞
), |XY |
∞
≤|X|
∞
·|Y |
∞
for all matrices U, V , X, Y with entries in C
∞
such that U + V and XY are
defined.
2.4. The ring E. We define E to be the ring consisting of formal power

series
∞

n=0
a
n
t
n
∈
¯
k[[ t]]
such that
lim
n→∞
n

|a
n
|
∞
=0, [k
∞
(a
0
,a
1
,a
2
, ):k
∞

] < ∞.
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
245
The former condition guarantees that such a series has an infinite radius of
convergence with respect to the valuation |·|
∞
. The latter condition guarantees
that for any t
0
∈ k
∞
the value of such a series at t = t
0
belongs again to k
∞
.
Note that the ring E is stable under the n-fold twisting operation f → f
(n)
for
all n ∈ Z.
2.5. The Schwarz-Jensen formula. Fix f ∈Enot vanishing identically.
It is possible to enumerate the zeroes of f in C
∞
because there are only finitely
many zeroes in each disc of finite radius. Put
{ω
i
} := an enumeration (with multiplicity) of the zeroes of f in C
∞
and

λ := the leading coefficient of the Maclaurin expansion of f.
The Schwarz-Jensen formula
sup
x∈
C
∞
|x|≤r
|f(x)|
∞
= |λ|
∞
· r
#{i|ω
i
=0}
·

i:0<|ω
i
|
∞
<r
r
|ω
i
|
∞
(r ∈ R
>0
)

relates the growth of the modulus of f to the distribution of the zeroes of f.
This fact is an easily deduced corollary to the Weierstrass Preparation Theorem
over a complete discrete valuation ring.
3. A linear independence criterion
3.1. Formulation and discussion of the criterion.
Theorem 3.1.1. Fix a matrix
Φ=Φ(t) ∈ Mat
×
(
¯
k[t]),
such that det Φ is a polynomial in t vanishing (if at all ) only at t = T . Fix a
(column) vector
ψ = ψ(t) ∈ Mat
×1
(E)
satisfying the functional equation
ψ
(−1)
=Φψ.
Evaluate ψ at t = T , thus obtaining a (column) vector
ψ(T ) ∈ Mat
×1

k
∞

.
For every (row ) vector
ρ ∈ Mat

1×
(
¯
k)
such that
ρψ(T )=0
246 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
there exists a (row ) vector
P = P(t) ∈ Mat
1×
(
¯
k[t])
such that
P (T )=ρ, Pψ =0.
The proof commences in §3.3 and takes up the rest of Section 3. We think
of the
¯
k[t]-linear relation P among the entries of ψ produced by the theorem
as an “explanation” or a “lifting” of the given
¯
k-linear relation ρ among the
entries of ψ(T ).
3.1.2. The basic application. Consider the power series
Ω=Ω(t):=
˜
T
−q
∞


i=1

1 − t/T
q
i

∈ k
∞
(

T )[[t]] ⊂ C
∞
[[ t]] .
The power series Ω(t) has an infinite radius of convergence and satisfies the
functional equation
Ω
(−1)
=(t −T ) ·Ω.
Consider the Maclaurin expansion
Ω(t)=
∞

i=0
a
i
t
i
.
The functional equation satisfied by Ω implies the recursion
q

√
a
i
+ Ta
i
=

a
i−1
if i>0,
0ifi =0.
Therefore Ω belongs to
¯
k[[ t]] and hence to E. Suppose now that there exists a
nontrivial
¯
k-linear relation
n

i=0
ρ
i
Ω(T )
i
=0 (ρ
i
∈
¯
k, n > 0,ρ
0

ρ
n
=0)
among the powers of the number
Ω(T )=
˜
T
−q
∞

i=1
(1 − T
1−q
i
) ∈ k
∞
(

T ).
Theorem 3.1.1 provides a
¯
k[t]-linear “explanation”
n

i=0
P
i
Ω
i
=0 (P

i
∈
¯
k[t],P
i
(T )=ρ
i
).
But the polynomial P
0
must vanish at all the zeroes t = T
q
i
of the function Ω.
Thus P
0
vanishes identically, contrary to our assumption that ρ
0
= P
0
(T ) =0.
We conclude that Ω(T ) is transcendental over k.
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
247
See §5.1 below for the interpretation of −1/Ω(T ) as the fundamental pe-
riod of the Carlitz module. The power series Ω(t) plays a key role in this
paper.
Proposition 3.1.3. Suppose
Φ ∈ Mat
×

(
¯
k[t]),ψ∈ Mat
×1
(C
∞
{t})
such that
det Φ(0) =0,ψ
(−1)
=Φψ.
Then
ψ ∈ Mat
×1
(E).
The proposition simplifies the task of checking the hypotheses of Theo-
rem 3.1.1.
Proof. Write
Φ=
N

i=0
b
(i)
t
i
(b
(i)
∈ Mat
×

(
¯
k),N: positive integer).
By hypothesis b
(0)
∈ GL

(
¯
k). By the theory of Lang isogenies [La a] there
exists U ∈ GL
×
(
¯
k) such that
U
(−1)
b
(0)
U
−1
= 1


equivalently: b
(1)
(0)
= U
−1
U

(1)

.
After making the replacements
ψ ← Uψ, Φ ← U
(−1)
ΦU
−1
,
we may assume without loss of generality that b
(0)
= 1

. Now write
ψ =
∞

i=0
a
(i)
t
i
(a
(i)
∈ Mat
×1
(C
∞
)).
We have

a
(−1)
(n)
− a
(n)
=
min(n,N)

i=1
b
(i)
a
(n−i)
,
and hence
a
(n)
∈ Mat
×1
(
¯
k)
for all integers n ≥ 0. By hypothesis
lim
n→∞
|a
(n)
|
∞
=0,

and hence the series
˜a
(n)
:=
∞

ν=1

N

i=1
b
(i)
a
(n−i)

(ν)
248 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
converges for all n  0. Moreover,
lim
n→∞
|˜a
(n)
|
∞
=0.
Since

˜a
(n)

− a
(n)

(−1)
=

˜a
(n)
− a
(n)

for n  0, it follows that ˜a
(n)
−a
(n)
= 0 for n  0 and hence that the collection
of entries of all the vectors a
(n)
generates an extension of k
∞
of finite degree.
Now fix C>1 arbitrarily. From the fact that ˜a
(n)
= a
(n)
for n  0, we have
inequalities
C
n
|a

(n)
|
∞
≤
N
max
i=1
C
n
|b
(i)
|
q
∞
|a
(n−i)
|
q
∞
≤

N
max
i=1
C
i
|b
(i)
|
q

∞

·

N
max
i=1
|a
(n−i)
|
∞

q−1
·

N
max
i=1
C
n−i
|a
(n−i)
|
∞

≤
n−1
max
i=0
C

i
|a
(i)
|
∞
for n  0, and hence
∞
sup
n=0
C
n
|a
(n)
|
∞
< ∞.
Therefore the radius of convergence of each entry of ψ is infinite.
3.1.4. Remark. Theorem 3.1.1 is in essence the (dual) t-motivic translation
of Yu’s Theorem of the t-Submodule [Yu c, Thms. 3.3 and 3.4]. Once the
setting is sufficiently developed, we expect that the class of numbers about
which Theorem 3.1.1 provides
¯
k-linear independence information is essentially
the same as that handled by Yu’s theorem of the t-Submodule, and the type
of information provided is essentially the same, too. We omit discussion of the
comparison.
3.2. Specialized notation for making estimates.
3.2.1. Degree in t. Given a polynomial f ∈
¯
k[t] let deg

t
f denote its
degree in t (as usual deg 0 := −∞) and, more generally, given a matrix F with
entries in
¯
k[t] put deg
t
F := max
ij
deg
t
F
ij
. Now, deg
t
F
(n)
= deg
t
F for all
n ∈ Z and we have
deg
t
(D + E) ≤ max (deg
t
D, deg
t
E) , deg
t
(FG) ≤ deg

t
F + deg
t
G
for all matrices D, E, F , G with entries in
¯
k[t] such that D + E and FG are
defined.
3.2.2. Size. Given an algebraic number x ∈
¯
k we set x := max
τ
|τx|
∞
,
where τ ranges over the automorphisms of
¯
k/k, thereby defining the size of x.
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
249
More generally given a polynomial f =

i
a
i
t
i
∈
¯
k[t], we define f  :=

max
i
a
i
. Yet more generally, given a matrix F with entries in
¯
k[t] we define
F  := max
ij
F
ij
. Then we have


F
(n)


= F 
q
n
for all n ∈ Z.Now,
D + E≤max(D, E), FG≤F·G
for all matrices D, E, F , G with entries in
¯
k[t] such that D + E and FG are
defined.
3.3. The basic estimates.
3.3.1. The setting. Throughout §3.3 we fix fields
k ⊂ K

0
⊂ K ⊂
¯
k
and rings
A ⊂O
0
⊂O⊂K
such that
• K
0
/k is a finite separable extension,
• K is the closure of K
0
in
¯
k under the extraction of q
th
roots,
•Ois the integral closure of A in K, and
•O
0
= O∩K
0
.
Note that
¯
k is the union of all its subfields of the form K.
3.3.2. Lower bound from size. We claim that
x≥1, |x|

∞
≥x
1−[K
0
:k]
for all 0 = x ∈O. Clearly these estimates hold in the case 0 = x ∈O
0
, because
in that case x has at most [K
0
: k] conjugates over k and the product of those
conjugates is a nonzero element of A; but then, since we have
O =
∞

ν=0
O
q
−ν
0
,
the claim holds in general.
Lemma 3.3.3 (Liouville Inequality). Fix a polynomial
f(z):=
n

i=0
a
i
z

i
∈O[z]
not vanishing identically. For every nonzero root λ ∈
¯
k of f (z) of order ν,
|λ|
ν
∞
≥

n
max
i=0
a
i


−[K
0
:k]
.
250 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
Proof. We may of course assume that |λ|
∞
< 1, for otherwise the claim
is obvious. After factoring out a power of z we may also assume that a
0
=0.
Write
f(z + λ)=

n

i=ν
b
i
z
i
(b
i
∈O[λ]),
noting that
|b
i
|
∞
≤
n
max
j=i
|a
j
|
∞
≤
n
max
i=0
a
i
.

Evaluate the displayed expression for f(z + λ)atz = −λ, thus obtaining an
estimate
|a
0
|
∞
= |f(0)|
∞
≤
n
max
i=ν
|b
i
λ
i
|
∞
≤|λ|
ν
∞
n
max
i=0
a
i
.
Finally, apply the fundamental lower bound of §3.3.2 to a
0
.

Lemma 3.3.4. For all constants C>1,
lim
ν→∞

#

x ∈O
q
−ν
0



x≤C

1
q
ν
·[K
0
:k]
= C.
The normalization |T |
∞
= q was imposed to make this formula hold.
Proof. We may assume without loss of generality that C is of the form
C = q
δ

δ ∈

∞

ν=0
q
−ν
Z,δ>0

.
The Riemann-Roch theorem yields constants n
0
and n
1
such that
n>n
0
⇒ #{x ∈O
0
|x≤|T|
n
∞
} = q
[K
0
:k]n+n
1
for all n ∈ Z. We then have
#

x ∈O
q

−ν
0



x≤|T|
δ
∞

=#

x ∈O
0



x≤|T|
q
ν
δ
∞

= q
[K
0
:k]q
ν
δ+n
1
for all integers ν  0, whence the result.

Lemma 3.3.5 (Thue-Siegel Analogue). Fix parameters
C>1, 0 <r<s (C ∈ R,r,s∈ Z).
For each matrix
M ∈ Mat
r×s
(O)
such that
M <C
there exists
x ∈ Mat
s×1
(O)
such that
x =0,Mx=0, x <C
r
s−r
.
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
251
Proof. Choose C

> 1 and ε>0 such that
M <C

, (1 + ε)(C

)
r
s−r
<C

r
s−r
.
For all ν  0 the cardinality of the set

x ∈ Mat
s×1

O
q
−ν
0




x≤(1 + ε)(C

)
r
s−r

exceeds the cardinality of the set

x ∈ Mat
r×1

O
q
−ν

0




x≤(1 + ε)(C

)
s
s−r

by Lemma 3.3.4. Further, for all ν  0 multiplication by M maps the former
set to the latter. Therefore the desired vector x exists by the pigeonhole
principle.
Lemma 3.3.6. Again fix parameters
C>1, 0 <r<s (C ∈ R,r,s∈ Z).
For each matrix
M ∈ Mat
r×s
(O[t])
such that
M <C
there exists
x ∈ Mat
s×1
(O[t])
such that
x =0,Mx=0, x <C
r
s−r

.
Proof. Let d and e be nonnegative integers presently to be chosen effica-
ciously large and put
r

:= r(d + e +1),s

:= s(e +1).
Choose d large enough so that
deg
t
M ≤ d,
and then choose e large enough so that
r

<s

, M <C

:= C
r
s−r
/
r

s

−r

.

Consider now the O-linear map
{x ∈ Mat
s×1
(O[t])|deg
t
x ≤ e}→{x ∈ Mat
r×1
(O[t])|deg
t
x ≤ d + e}
induced by multiplication by M. With respect to the evident choice of bases
the map under consideration is represented by a matrix
M

∈ Mat
r

×s

(O)
252 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
such that


M



<C


.
The existence of x ∈ Mat
s×1
(O[t]) such that
x =0,Mx=0, deg
t
x ≤ e, x < (C

)
r

s

−r

= C
r
s−r
now follows by an application of the preceding lemma with the triple of pa-
rameters (C

,r

,s

) in place of the triple (C, r, s).
3.4. Proof of the criterion.
3.4.1. The case  = 1. Assume for the moment that  = 1. In this case
we may assume without loss of generality that ρ = 0 and hence that
ψ(T )=0,

in which case our task is to show that ψ vanishes identically. For any integer
ν ≥ 0 we have

ψ

T
q
−ν

q
−1
= ψ
(−1)

T
q
−(ν+1)

=Φ

T
q
−(ν+1)

ψ

T
q
−(ν+1)


,
Φ

T
q
−(ν+1)

=0
and hence,
ψ

T
q
−ν

=0 (ν =0, 1, 2, ).
Since ψ vanishes infinitely many times in the disc |t|
∞
≤|T|
∞
, necessarily ψ
vanishes identically. Thus the case  = 1 of Theorem 3.1.1 is dispatched.
3.4.2. Reductions and further notation. Assume now that >1. We
may of course assume that
ρ =0.
As in §3.3 let
k ⊂ K
0
⊂ K ⊂
¯

k
be fields such that K
0
/k is a finite separable extension and K is the closure of
K
0
under the extraction of q
th
roots. Since
¯
k is the union of fields of the form
K we may assume without loss of generality that
Φ ∈ Mat
×
(K[t]),ρ∈ Mat
1×
(K).
As in §3.3 let O be the integral closure of A in K. After making replacements
Φ ← a
q−1
Φ,ψ← a
−q
ψ, ρ ← bρ
for suitably chosen
a, b ∈ A, ab =0,
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
253
we may assume without loss of generality that
Φ ∈ Mat
×

(O[t]),ρ∈ Mat
1×
(O).
Fix a matrix
ϑ ∈ Mat
×(−1)
(O)
of maximal rank such that
ρϑ =0.
Then the K-subspace of Mat
1×
(K) annihilated by right multiplication by ϑ
is the K-span of ρ. Let
Θ ∈ Mat
×
(O[t])
be the transpose of the matrix of cofactors of Φ. Then,
ΦΘ = ΘΦ = det Φ ·1

= c(t − T )
s
· 1

for some 0 = c ∈Oand integer s ≥ 0. Let N be a parameter taking values in
the set of positive integers divisible by 2.
3.4.3. Construction of the auxiliary function E. We claim there exists
h = h(t) ∈ Mat
1×
(O[t])
depending on the parameter N such that

•h = O(1) as N →∞
and with the following properties for each value of N:
• h =0.
• deg
t
h<

1 −
1
2

N.
• E

T
q
−(N+ν)

= 0 for ν =0, ,N − 1, where E := hψ ∈E.
(We call E the auxiliary function.)
Before proving the claim, we note first that the auxiliary function E figures in
the following identity:
hΘ
(−0)
···Θ
(−(N+ν−1))
ψ
(−(N+ν))
= hΘ
(−0)

···Θ
(−(N+ν−1))
Φ
(−(N+ν−1))
···Φ
(−0)
ψ
= c
q
−(N+ν−1)
+···+q
0

t − T
q
−(N+ν−1)

s
···

t − T
q
−0

s
E.
254 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
This identity is useful again below and so for convenient reference we dub it
the key identity. By the key identity, the hypothesis
ρψ(T )=0


equivalently: ρ
(−(N+ν))
ψ
(−(N+ν))

T
q
−(N+ν)

=0

,
and the definition of ϑ, the following condition forces the desired vanishing of
E:
• hΘ
(0)
···Θ
(−(N+ν−1))
ϑ
(−(N+ν))


t=T
q
−(N+ν)
= 0 for ν =0, ,N − 1.
Put
r := ( − 1) N, s :=


 −
1
2

N.
With respect to the evident choices of bases, the homogeneous system of linear
equations that we need to solve is described by a matrix
M ∈ Mat
r×s
(O)
depending on N such that
M≤|T |
q
−N
((1−
1
2
)N+2N·deg
t
Θ)
∞
·Θ
q
q−1
·ϑ = O(1) as N →∞,
and the solution we need to find is described by a vector
x ∈ Mat
s×1
(O)
depending on N such that

x =0,Mx=0, x = O(1) as N →∞.
Lemma 3.3.5 now proves our claim.
3.4.4. A functional equation for E. We claim there exist polynomials
a
0
, ,a

∈O[t]
depending on the parameter N such that
•

max
i=0
a
i
 = O(1) as N →∞
and with the following properties for each value of N:
• Not all the a
i
vanish identically.
• a
0
E + a
1
E
(−1)
+ ···+ a

E
(−)

=0.
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
255
Since
E
(−ν)
= h
(−ν)
Φ
(−(ν−1))
···Φ
(−0)
ψ
for any integer ν ≥ 0, the functional equation we want E to satisfy is implied
by the following condition:
• a
0
h
(0)
+ a
1
h
(−1)
Φ
(−0)
+ ···+ a

h
(−)
Φ

(−(−1))
···Φ
(−0)
=0.
The latter system of homogeneous linear equations for a
0
, ,a

is with respect
to the evident choice of bases described by a matrix
M ∈ Mat
×(+1)
(O[t])
depending on N such that
M = O(1) as N →∞,
and the solution we have to find is described by a vector
x ∈ Mat
(+1)×1
(O[t])
depending on N such that
x =0,Mx=0, x = O(1) as N →∞.
Lemma 3.3.6 now proves our claim. After dividing out common factors of t we
may further assume that for each value of N:
• Not all the constant terms a
i
(0) vanish.
3.4.5. Vanishing of E. We claim that E vanishes identically for some N.
Suppose that this is not the case. Let λ be the leading coefficient of the
Maclaurin expansion of E. We have
a

0
(0)λ
q
0
+ ···+ a

(0)λ
q
−
=0,
and hence
1/|λ|
∞
= O(1) as N →∞
by Lemma 3.3.3. But we also have
|λ|
∞
·|T |
N−
q
q−1
∞
≤ sup
x∈
C
∞
|x|
∞
≤|T |
∞

|E(x)|
∞
≤ sup
x∈
C
∞
|x|
∞
≤|T |
∞
|ψ(x)|
∞
·h·|T |
N
(
1−
1
2
)
∞
,
for all N , the inequality on the left by the Schwarz-Jensen formula, and hence
|λ|
∞
= O

|T |
−
N
2

∞

as N →∞.
These bounds for |λ|
∞
are contradictory for N  0. Our claim is proved.
256 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
3.4.6. The case E = 0. Now fix a value of N such that the auxiliary func-
tion E vanishes identically. Since the entries of the vector h are polynomials
in t of degree <N, not all vanishing identically, there exists some 0 ≤ ν<N
such that
h
(N+ν)
(T )=h

T
q
−(N+ν)

q
N+ν
=0.
Put
P = P(t):=h
(N+ν)
Θ
(N+ν)
···Θ
(1)
∈ Mat

1×
(O[t]).
Since
det

Θ
(N+ν)
···Θ
(1)




t=T
=0,
we have
P (T ) =0.
We also have
P (T )ϑ =

hΘ
(−0)
···Θ
(−(N+ν−1))
ϑ
(−(N+ν))



t=T

q
−(N+ν)

q
N+ν
=0,
and hence
P (T ) ∈ (K-span of ρ) ⊂ Mat
1×
(K).
Finally, we have
Pψ = c
q+···+q
N+ν
(t − T
q
)
s
···

t − T
q
N+ν

s
E
(N+ν)
=0
by the key identity. Therefore (up to a nonzero correction factor in K) the
vector P is the vector we want, and the proof of Theorem 3.1.1 is complete.

4. Tools from (non)commutative algebra
4.1. The ring
¯
k[σ].
4.1.1. Definition. Let
¯
k[σ] be the ring obtained by adjoining a noncom-
mutative variable σ to
¯
k subject to the commutation relations
σx = x
q
−1
σ (x ∈
¯
k).
Every element of
¯
k[σ] has a unique presentation of the form
∞

i=0
a
i
σ
i
(a
i
∈
¯

k, a
i
= 0 for i  0),
and in terms of such presentations the multiplication law in
¯
k[σ] takes the form


i
a
i
σ
i



j
b
j
σ
j

=

i

j
a
i
b

q
−i
j
σ
i+j
.
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
257
Given
φ =
∞

i=0
a
i
σ
i
∈
¯
k[σ](a
i
∈
¯
k, a
i
= 0 for i  0),
we define
deg
σ
φ := max({−∞} ∪ {i|a

i
=0}).
Clearly we have
deg
σ
φψ = deg
σ
φ + deg
σ
ψ (φ, ψ ∈
¯
k[σ]).
The ring
¯
k[σ] admits interpretation as the opposite of the ring of F
q
-linear
endomorphisms of the additive group over
¯
k. This interpretation is not actually
needed in the sequel but might serve as a guide to the intuition of the reader.
4.1.2. Division algorithms and their uses. The ring
¯
k[σ] has a left (resp.,
right) division algorithm:
• For all ψ,φ ∈
¯
k[σ] such that φ = 0 there exist unique θ,ρ ∈
¯
k[σ] such

that ψ = φθ + ρ (resp., ψ = θφ + ρ) and deg
σ
ρ<deg
σ
φ.
Some especially useful properties of
¯
k[σ] and of left modules over it readily
deducible from the existence of left and right division algorithms are as follows:
• Every left ideal of
¯
k[σ] is principal.
• Every finitely generated left
¯
k[σ]-module is noetherian.
• dim
¯
k
¯
k[σ]/
¯
k[σ]φ = deg
σ
φ<∞ for all 0 = φ ∈
¯
k[σ].
• For every matrix φ ∈ Mat
r×s
(
¯

k[σ]) there exist matrices α ∈ GL
r
(
¯
k[σ])
and β ∈ GL
s
(
¯
k[σ]) such that the product αφβ vanishes off the main
diagonal.
• A finitely generated free left
¯
k[σ]-module has a well-defined rank; i.e., all
¯
k[σ]-bases have the same cardinality.
• A
¯
k[σ]-submodule of a free left
¯
k[σ]-module of rank s<∞ is free of rank
≤ s.
• Every finitely generated left
¯
k[σ]-module is isomorphic to a finite direct
sum of cyclic left
¯
k[σ]-modules.
These facts are quite well known. The proofs run along lines very similar to
the proofs of the analogous statements for, say, the commutative ring

¯
k[t].
258 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
4.1.3. The functors mod σ and mod (σ − 1). Given a homomorphism
f : H
0
→ H
1
of left
¯
k[σ]-modules, let
f mod σ :
H
0
σH
0
→
H
1
σH
1
,fmod (σ −1) :
H
0
(σ −1)H
0
→
H
1
(σ −1)H

1
be the corresponding induced maps.
Lemma 4.1.4. Let
f : H
0
→ H
1
be an injective homomorphism of free left
¯
k[σ]-modules of finite rank such that
n := dim
¯
k
coker(f ) < ∞.
Now,
# ker(f mod (σ − 1)) ≤ q
n
with equality if and only if f mod σ is bijective.
Proof. We may assume without loss of generality that
H
0
= Mat
1×r
(
¯
k[σ]),H
1
= Mat
1×s
(

¯
k[σ]),f=(x → xφ)(φ ∈ Mat
r×s
(
¯
k[σ])).
After replacing φ by αφβ for suitably chosen α ∈ GL
r
(
¯
k[σ]) and β ∈ GL
s
(
¯
k[σ]),
we may assume without loss of generality that φ vanishes off the main diagonal,
in which case clearly φ vanishes nowhere on the main diagonal and r = s.We
might as well assume now also that r = s = 1. Write
φ =
n

i=0
a
i
σ
i
(a
i
∈
¯

k, a
n
=0,a
0
=0⇔ f mod σ is bijective).
Now,
¯
k[σ]=
¯
k ⊕(σ − 1) ·
¯
k[σ]
and
xφ ≡
n

i=0
a
q
i
i
x
q
i
mod (σ −1) ·
¯
k[σ]
for all x ∈
¯
k, whence the result.

Lemma 4.1.5. For i =1, 2 let
f
i
: H
0
→ H
i
be a homomorphism of free, left
¯
k[σ]-modules of finite rank. Assume that H
0
,
H
1
and H
2
are all of the same rank over
¯
k[σ]. Assume further that f
1
mod σ
is bijective and that
ker(f
1
mod (σ −1)) ⊂ ker(f
2
mod (σ −1)).
ALGEBRAIC RELATIONS AMONG Γ-VALUES IN CHARACTERISTIC p
259
Then f

2
factors uniquely through f
1
; i.e., there exists a unique homomorphism
g : H
1
→ H
2
of left
¯
k[σ]-modules such that
g ◦f
1
= f
2
.
Proof. (Cf. [Yu c, Lemma 1.1].) We may assume without loss of generality
that
H
0
= H
1
= H
2
= Mat
1×s
(
¯
k[σ]),
f

1
=(x → xφ),f
2
=(x → xψ), (φ, ψ ∈ Mat
s×s
(
¯
k[σ])).
After making replacements
φ ← αφβ, ψ ← αψ
for suitably chosen α, β ∈ GL
s
(
¯
k[σ]), we may assume without loss of generality
that φ vanishes off the main diagonal. Since f
1
mod σ is bijective, no diagonal
entry of φ vanishes. We might as well assume now also that s = 1. Use the
left division algorithm to find θ, ρ ∈
¯
k[σ] such that
ψ = φθ + ρ, deg
σ
ρ<deg
σ
φ.
Put
g := (x → xθ),h:= (x → xρ).
Then

f
2
= g ◦f
1
+ h.
If h = 0 we are done. Suppose instead that h = 0. We then have
ker(f
1
mod (σ −1)) ⊂ ker(h mod (σ − 1)), dim
¯
k
coker(f
1
) > dim
¯
k
coker(h).
But the latter relations are contradictory in view of Lemma 4.1.4 and our
hypothesis that f
1
mod σ is bijective.
4.2. The ring
¯
k[[ σ]] .
4.2.1. Definition. We define
¯
k[[ σ]] to be the completion of
¯
k[σ] with
respect to the system of two-sided ideals


σ
n
¯
k[σ]

∞
n=0
.
Every element of
¯
k[[ σ]] has a unique presentation of the form
∞

i=0
a
i
σ
i
(a
i
∈
¯
k).
In terms of such presentations the multiplication law in
¯
k[[ σ]] takes the form


i

a
i
σ
i



j
b
j
σ
j

=

i

j
a
i
b
q
−i
j
σ
i+j
.
260 G. W. ANDERSON, W. D. BROWNAWELL, AND M. A. PAPANIKOLAS
The ring
¯

k[[ σ]] contains
¯
k[σ] as a subring. The ring
¯
k[[ σ]] is a domain.
4.2.2. The operation ∂. Given
φ =
∞

i=0
a
(i)
σ
i
∈ Mat
r×s
(
¯
k[[ σ]] ) ( a
(i)
∈ Mat
r×s
(
¯
k)),
put
∂φ := a
(0)
.
The operation ∂ thus defined is

¯
k-linear and satisfies
∂(φψ)=(∂φ)(∂ψ)
for all matrices φ and ψ with entries in
¯
k[[ σ]] such that the product φψ is
defined.
Lemma 4.2.3. (i) For al l φ ∈ Mat
s×s
(
¯
k[[ σ]]), if ∂φ ∈ GL
s
(
¯
k), then φ ∈
GL
s
(
¯
k[[ σ]] ) . (ii) Every nonzero left ideal of
¯
k[[ σ]] is generated by a power of σ.
Proof. (i) After replacing φ by αφ for suitably chosen α ∈ GL
s
(
¯
k)wemay
assume ∂φ = 1
s

. Now write φ = 1
s
−X. The series 1
s
+

∞
n=1
X
n
converges to
a two-sided inverse to φ. (ii) Let I ⊂
¯
k[[ σ]] be a nonzero left ideal. Let φ = ασ
n
be a nonzero element of I where ∂α = 0 and n is a nonnegative integer taken
as small as possible. Then we have α ∈
¯
k[[ σ]]
×
by (i); hence σ
n
∈ I, and hence
σ
n
generates I.
Lemma 4.2.4. Let
θ ∈ Mat
r×r
(

¯
k[[ σ]] ) ,a∈ Mat
r×r
(
¯
k),e∈ Mat
r×s
(
¯
k),b∈ Mat
s×s
(
¯
k)
be given such that
∂θ = a, (a − T · 1
r
)
r
=0,ae= eb, (b − T · 1
s
)
s
=0.
Then there exists unique
E ∈ Mat
r×s
(
¯
k[[ σ]] )

such that
θE = Eb, ∂E = e.
Proof. (Cf. [An a, Prop. 2.1.4].) Write
θ =
∞

i=0
a
(i)
σ
i
∈ Mat
r×r
(
¯
k[[ σ]] ) ( a
(i)
∈ Mat
r×r
(
¯
k),a
(0)
= a)
and
E =
∞

i=0
e

(i)
σ
i
∈ Mat
r×s
(
¯
k[[ σ]] ) ( e
(i)
∈ Mat
r×s
(
¯
k),e
(0)
= e).